You may have noticed that we have cheated a bit. In Chapter 1 ‘‘Statistics for Business,’’ we defined statistics as the use of numbers to describe general facts about the world. Now, we have shown how some measurements used in statistics are not really numbers at all, at least not in the ordinary sense that we learned about numbers in high school. Statistics uses an expanded notion of number that includes other sorts of symbol systems. The statistical notion of number does have its limits. First of all, the non-numeric values used in statistics must be part of a formal system that can be treated mathematically. In this section, we will learn about the most common systems used in statistics. Also, for most statistical techniques used in inferential statistics, the values will need to be converted into numbers, because inferential statistical techniques use algebra, which requires numbers.
Let’s start with our example of measuring Judy’s height. We say that that measurement results in a number, 62. You may remember from high school algebra (or else from Appendix A) that there is more than just one kind of number. There are counting numbers, integers, rational numbers, real numbers, and so forth. We will see that it matters a lot what kind of number
we use for different kinds of measurements. Height is measured with positive
real numbers. A person can be 5 foot 101
2inches tall, but they can’t be minus
six feet tall, or zero inches tall.
We can see that the type of number used for different kinds of measurement depends on what different values are possible outcomes of that type of measurement. The number of items on a receipt is measured as a positive integer, also known as a counting number. Counting numbers are non-negative integers because counts don’t include fractions (ordinarily) or negative values. The number of children in a family could be zero (technically, a non-negative integer). A bank balance, whether measured in dollars or in cents, is an integer, because it can be negative as well as positive (negative if there is an overdraft), but we can’t have fractions of pennies. Height and weight are positive real numbers. The amount of oil in an oil tanker could be zero as well as a positive value. So it is measured as a non- negative real number. The temperature inside a refrigerated container could be negative or positive or zero, at least in the Celsius or Fahrenheit scales.
KEY POINT
In algebra, different types of numbers are defined in terms of the different possible values included. We choose the type of number for measuring a particular type of variable when the different possible numeric values match up to the different measurement outcomes.
But what about measurements that don’t result in numbers? Let’s go back to our example of making an inventory of cars in a car lot. Suppose that each parking spot in the car lot is labeled from A to Z. Each car is either a sedan, convertible, or minivan. Our inventory sheet, shown in Table 2-1, has one line for each parking spot on the lot. We go through the lot and write down the model of the car in the line corresponding to its parking spot.
Car models, like height, or weight, or dollars in a bank account, have different values for different subjects, but the different values don’t really correspond well to the different values for different types of numbers. The closest match is positive integers, by assigning different numbers to different models, like 1 for sedan, 2 for convertible, and 3 for minivan, but there is a problem with this as well.
Integers are different from car models in two ways. The first problem is minor. There are an infinite number of integers, but only a finite number of car models. Every bank account may have a finite amount of money in it, but in principle, there is no limit to how much money can be in our bank account. That is a good reason to use integers to measure money. Similarly, new car models, like the minivan, occasionally get invented, so the infinite number of integers available may be handy.
The other problem is not so minor. The integers possess a very important property that car models do not: the property of order. Three is bigger than two, which is bigger than one. There is no relation like ‘‘bigger than’’ that applies to car models. The best way to see this is to realize that there is no reason to choose any particular number for any particular car model. Instead of choosing 1 for sedan, 2 for convertible, and 3 for minivan, we could just as easily have chosen 1 for convertible, 2 for minivan, and 3 for sedan. Our
choice of which number to use isarbitrary.And arbitrary is not a good thing
when it comes to mathematics.
Statisticians do not classify different types of measurement in terms of what types of numbers (or non-numerical symbols) are used to record the results. While it may make a difference to certain types of calculations used in statistics as to whether the original measurements are integers or real numbers, this difference does not figure into the classification of measure- ment. Instead, they group the different types of numbers in terms of what
Table 2-1 Automobile inventory.
Parking spot Type of car
A sedan B sedan C convertible D sedan E minivan F minivan . . . .
makes a difference in using different statistical techniques. Just as with
statistical assumptions, the different types of measurement, called levels of
measurement, are grounded in the very important issue of how to pick the right sort of statistical analysis for the problem at hand. The different levels of measurement are:
. Nominal scale.When the values have no relation of order, the variable is said to be on a nominal scale. This corresponds to categorical data. Example: Methods of drug administration: oral, intravenous, intra- muscular, subcutaneous, inhalant, topical, etc.
. Ordinal scale. When the values have a relation of order, but intervals between adjacent values are not equal, the variable is said to be on an ordinal scale. This is one type of numerical data. Example: Coin grades: Poor, Fair, Good, Very Good, Fine, Very Fine, Extra Fine, Mint, etc.
. Interval scale. When the values have a relation of order, and intervals between adjacent values are equal, but a value of zero is arbitrary, the variable is said to be on an interval scale. This is another type of numerical data. Example: Fahrenheit temperature.
. Ratio scale. When the values have a relation of order, the intervals between adjacent values are equal, and a value of zero is meaningful, the variable is said to be on a ratio scale. (A meaningful value of zero
is called a true zero pointor origin.) This is the last type of numerical
data. Example: Money, with debt measured as negative numbers.
HANDY HINTS
Some textbooks define ordinal data as a form of categorical data and others as a form of numerical data. This is because ordinal data has characteristics of each and, depending on what we do with it, it may be treated as either. An ordinal variable does classify each individual subject item into one and only one category and, by that standard, is definitely a type of categorical variable, where the categories have a specific order. When graphing, ordinal variables are treated as categorical. Because the positive integers are a very convenient way of showing order (after all, we are all pretty familiar with the counting order), ordinal variables are very often coded numerically as positive integers, which is one reason why some textbooks classify ordinal variables as numerical.
Finally, many statistical inference techniques that require an interval level of measurement can be and are used effectively with ordinal variables coded as integers. (This is a good example of using a statistical technique even though one of itsassumptionsis violated.) When it comes to inferential statistics, ordinal variables
are treated as categorical or numerical depending on the technique used. Using a technique (called a nonparametric technique) designed for categorical variables will be more accurate, but may be less powerful. (That is, the technique is more likely to fail to give a definitive answer to our question.) Using a technique (called a parametric technique) designed for numerical variables is more powerful, but less accurate, because the fact that the adjacent categories of an ordinal variable are not guaranteed to be equally far apart violates one of the assumptions of the technique.
There is also a special case of a nominal variable that can be treated as interval. When a variable can take on only two values, like true and false, or male and female, or is-a-current-customer and is-not-a-current-customer, the data are nominal because there is no order to the values. When used in inferential statistics, these variables can be treated as interval, because, having only two possible values, they only have one interval between the values. And one interval is always equal to itself. Variables that can take on only two values are sometimes calledbinary variables, most often calleddichotomous variables, and when used in the inferential technique known as
regression(see Chapter 12 ‘‘Correlation and Regression’’), asdummy variables.
We will learn more about all of this in Part Three, where we learn about inferential statistical techniques.
Note that this classification system ignores the differences between integers, rational numbers, and real numbers. This is because measurements are always made up to some level of precision. There is always the possibility that two values are so close that they cannot be distinguished. Two people, where one is six feet tall and the other is six feet and one millionth of an inch tall, will both be classified as six feet tall. For the purpose of the analysis, there is no difference between them. There are no truly continuous numbers in measurement. Since statistics always begins with measurement, the issue of continuity is irrelevant in applied statistics.
The only exception to this rule is for measurements that don’t ever come in fractions. For example, sometimes the general fact of the world we care about is discovered by counting, as in the number of widgets we produced last week. The number of widgets is always a whole number. It wouldn’t make
much sense to say we have 4512 widgets on hand. As we will see in later
chapters, statistics handles this problem in two different ways. If the number of items is large enough, many of our questions can be answered statistically by pretending that fractional values are possible. For example, if we are producing between 40 and 50 thousand widgets a month, the fact that the detailed calculations use fictitious values like 42,893.087 instead of genuinely possible values like 42,893, doesn’t matter much. If the number of items is small (usually less than 20), and it is the count that we really care about, there
questions. In order to keep this difference straight, we will have two separate examples running through the book: one about counting sheep, and one about measuring people.
As we will see later on in Part Two and Part Three, the issues of possessing order, equal intervals, and a true zero point are used to classify variables because they make a difference as to whether different statistical measures and techniques can be used effectively.
Error
In order to help make decisions, we need to know the true value of the information that statistics provides. Statistics not only provides information, but also specific measures of the degree of confidence with which that information can be trusted. This ability to measure the quality of statistical information is based on the concept of error.
TIPS ON TERMS
Error.The degree to which a description does not match whatever is being described.
All aspects of statistics are prone to error. No individual measurement is free from error. Measurement is a human process, limited by our tools and our senses and our other fallible human capacities. We need to understand measurement error in order to have the right amount of confidence in our data. Statistical measures and statistical techniques are also prone to error of another type. Even when calculated mechanically and exactly from the data, the information statistics gives us is never an exact description of the true state of the world. (We will see more of why this is so later on in this chapter and also in Chapter 3 ‘‘What Is Probability?’’) The statistical theory of error helps us gauge the right amount of confidence to have in both our data and our statistics.